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F Numerical issues in dynamic heat exchanger models. As stated in Eq. 109, chattering and simulations failures are likely if: ➢ The number of cells is low (Figure 103) ➢ The cycle working fluid flow rate is low ➢ The heat exchanger internal volume is high ➢ The working conditions are highly transient (i.e. dhi/dt and dp/dt are high) In order to compare the efficiencies of different numerical methods aiming at addressing this issue, a so-called "stressed" system is defined: this system is a heat exchanger (evaporator) submitted to very transient operating conditions and to a low working fluid flow rate. The following boundary conditions are defined: ➢ Heat source: Water, 0.15 kg/s, 200°C ➢ Supply conditions: HFC-245fa, 0.37 kg/s, 50°C ➢ Heat exchanger parameters: N=20, V=3l (For the other parameters, see Figure 60) ➢ Evaporating pressure: p=12+5⋅sin(0.2⋅π⋅t) [bar] The internal heat exchanger volume of 3l is selected by increasing its value until reaching a simulation failure. The highly transient character of the operating conditions is ensured by a sine wave on the pressure, with an amplitude of 5 bars and a frequency of 0.1 Hz. The period of 10s is smaller than the natural response time of the heat exchanger, resulting in pressure and enthalpy derivatives much higher than in traditional ORC simulations. Robustness strategies Constant node flow rates. This strategy avoids the generation of a high node flow rate in one particular cell by applying the mass conservation equation between the inlet and the outlet of the whole heat exchanger instead of every cell. The mass variations inside the cells are summed and reported on the last node flow rate. This is written: M ̇ *i =M ̇ su (110) N ∂ρidhi ∂ρidp N dρi M ̇ex=M ̇su−Vi⋅∑1 dt=M ̇su−Vi⋅∑1 [∂h⋅dt+∂p⋅dt] (111) Maximum density derivative. In this strategy, the peak in the density derivative occurring after the transition from liquid to two-phase is truncated, as shown in Figure 104. This strategy allows conserving the mass balance equation in each cell. The simulation is therefore not affected, except when a phase transition occurs. In this case, the mass balance is not respected and a mass default can appear. Figure 104 shows however that the truncated area is relatively small, which should reduce the mass unbalance. XXPDF Image | Organic Rankine Cycles for Waste Heat Recovery and Solar Uses
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